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Proof step 44 from proof congmod-def-mod-02 (previous step)

0 ≤ i % j, i < 0, i0 < 0, i0 % j < 0 ⊦ j = 0, i % j = i0 % j + abs(j) ↔ i % j = i0 % j + j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: Abs-pos seq: ⊦ ¬ j < 0 → abs(j) = j subst: [j] → [j] flags: rewrite

Additional information

Used simplifier rules: abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%neg-15: ⊦ i + ~ j = i - j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%mod-ngez-01: ⊦ i < 0 ∧ j < 0 → (0 ≤ i % j ↔ i % j = 0); from specification int-div%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%ax-04: ⊦ i + 0 = i; from specification int-basic2%mod-neq: j ≠ 0 ⊦ i % j ≠ j; from specification int-div%zero: ⊦ i < 0 ∧ j < 0 → i + j ≠ 0; from specification int-basic2 Used terms: j